Examples of point groups and their characters – more details PHY 745 Group Theory 11-11:50 AM MWF Olin 102 Plan for Lecture 8: Examples of point groups and their characters – more details Reading: Chapter 4 & 8 in DDJ More details on basis functions More details on symmetries of molecular vibrations Note: In this lecture, some materials are taken from an electronic version of the Dresselhaus, Dresselhaus, Jorio text 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

1/30/2017 PHY 745 Spring 2017 -- Lecture 8

Point symmetry groups in physics Notion of symmetry related basis functions* Properties: Orthogonality: *(Using notation in DDJ textbook) 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

Examples of basis functions based on Cartesian coordinates for the example of the triangular group: P(3)=D3 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

Summary of basis functions associated with character table for D3 “Standard” notation for representations of D3 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

Example basis functions: z y Comment on orthogonality: Assume integral implied by braket takes place over unit sphere and appropriate normalization constants will be applied. 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

z x y Note that basis functions related to irreducible representations follow: 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

z x y 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

1/30/2017 PHY 745 Spring 2017 -- Lecture 8

Extension to quadratic basis functions 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

Summary of basis functions associated with character table for D3 “Standard” notation for representations of D3 In this case, Rx, Ry, Rz behave as axial vectors along x, y, z. 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

“Standard” notation for representations of C2v Example of H2O sv sv’ “Standard” notation for representations of C2v 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

Basis functions for this case -- z C2 sv sv’ x y 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

Basis functions for C2v – continued -- 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

Vibrational modes From: B1 A1 A1 http://chem.libretexts.org/Core/Physical_and_Theoretical_Chemistry/Spectroscopy/Vibrational_Spectroscopy/Vibrational_Modes B1 A1 A1 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

Another example – CH4 (tetrahedral symmetry) http://www2.ess.ucla.edu/~schauble/MoleculeHTML/CH4_html/CH4_page.html (2 modes) T2 T2 (3 modes) (3 modes) 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

Symmetry analysis of vibrations of CH4 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

Visualization of symmetry elements http://symmetry.otterbein.edu/tutorial/methane.html sd C3 C2 S4 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

Dz1 Construct 15-component vector V describing the 3-dimensional motion of the 5 atoms in CH4 Dy1 Dx1 Dz5 Dz4 Dz2 Dy5 Dx5 Dz3 Dy4 Dy2 Dx4 Dx2 Dy3 Dx3 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

Alternate coordinate system Dz1 Dz1 Dz2 Dy1 Dx1 Dz5 Dx1 Dy1 Dx2 Dy2 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

All motions: A1+E+T1+3T2 Translations: T2 Rotations: T1 Vibrations: A1+E+2T2 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

Symmetry of NH3 C3v http://wps.prenhall.com/wps/media/objects/602/616516/Media_Assets/Chapter19/Text_Images/FG19_07-12UN.JPG 1/30/2017 PHY 745 Spring 2017 -- Lecture 8

1/30/2017 PHY 745 Spring 2017 -- Lecture 8